Ito’s formula

We will review Levy stochastic integrals and Ito’s formula.

Let ${X = M + B}$ be a semimartingale, where ${M}$ is a martingale and ${B}$ is a process of bounded variation. The problem of stochastic integration is to make sense of objects of the form

$\displaystyle \int_0^t F(s) d X(s) = \int_0^t F(s) d M(s) + \int_0^t F(s) d B(s).$

If ${X}$ is a Levy process, then there exists ${b\in \mathbb{R}^d}$, a Brownian motion ${W_A}$ with covariance matrix ${A}$ in ${\mathbb{R}^d}$, and an independent Poisson random measure ${N}$ on ${\mathbb{R}^+\times (\mathbb{R}^d \setminus \{0\})}$ with Levy measure ${\nu}$, such that for each ${t\ge 0}$, Levy-Ito decomposition has the form of

$\displaystyle X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx). \ \ \ \ \ (1)$

Given ${E \in \mathcal{B}(\mathbb{R}^d)}$, define

$\displaystyle \rho((s,t],E) = (t-s) (\delta_0(E) + \nu(E\setminus \{0\})).$

Let ${\mathcal{P}_2 (T, E)}$ (resp. ${\mathcal{H}_2(T,E)}$) be the linear space of all equivalence classes of mappings ${F: [0,T] \times E \times \Omega \rightarrow \mathbb{R}}$ which coincide almost everywhere with respect to ${\rho \times P}$, and satisfies the following conditions:

1. ${F}$ is predictable, (see definition here)
2. ${P(\int_0^T \int_E |F(t,x)|^2 \rho(dt, dx) <\infty) = 1.}$ (resp. ${\int_0^T \int_E \mathbb{E}[ |F(t,x)|^2] \rho(dt, dx) <\infty.}$)

${\mathcal{P}_2(T)}$ (resp. ${\mathcal{H}_2(T)}$) is the space of maps ${F: [0,T] \times \Omega \rightarrow \mathbb{R}}$, which is predictable and ${P(\int_0^T |F(t)|^2 dt < \infty) = 1}$ (resp. ${\int_0^T \mathbb{E}[ |F(t)|^2] dt < \infty}$). Note that ${\mathcal{H}_2(T, E)}$ and ${\mathcal{H}_2(T)}$ are Hilbert spaces, and ${\mathcal{H}_2(T, E) \subset \mathcal{P}_2(T, E)}$ and ${\mathcal{H}_2(T) \subset \mathcal{P}_2(T).}$

In this below, we take ${E = B_1(0) \setminus \{0\}}$. We say that an ${\mathbb{R}^d}$-valued stochastic process ${Y = (Y(t), t\ge 0)}$ is a Levy-type stochastic integral if it can be written in the following form for each ${1\le i \le d}$, ${t\ge 0}$

$\displaystyle \begin{array}{l} Y^i(t) =\displaystyle Y^i(0) + \int_0^t G^i(s) ds + \sum_{j=1}^m \int_0^t F_j^i(s) d W^j(s) \\ \hspace{0.5in} \displaystyle+ \int_0^t \int_{|x|<1} H^i(s,x) \tilde N(ds, dx) + \int_0^t \int_{|x|\ge 1} K^i(s,x) N(ds, dx), \end{array} \ \ \ \ \ (2)$

where for each ${1\le i \le d, 1\le j \le m}$, ${t\ge 0}$, ${|G^i|^{1/2}, F_j^i\in \mathcal{P}_2(T)}$, ${H^i \in \mathcal{P}_2(T, E)}$, and ${K}$ is predictable. ${W}$ is an ${m}$-dimensional standard Brownian motion and ${N}$ is an independent Poisson random measure on ${\mathbb{R}^+ \times (\mathbb{R}^d\setminus \{0\})}$ with compensator ${\tilde N}$ and intensity measure ${\nu}$, which we will assume is a Levy measure.

Equivalent notation for Levy-type stochastic integrals of (2) is either

$\displaystyle d Y(t) = G(t) dt + F(t) dW(t) + H(t,x) \tilde N(dt, dx) + K(t,x) N(dt, dx). \ \ \ \ \ (3)$

or (to emphasize the domains of integration)

$\displaystyle d Y(t) = G(t) dt + F(t) d W(t) + \int_{|x|<1}H(t,x) \tilde N (dt, dx) + \int_{|x|\ge 1}K(t,x) N(dt, dx).$

Clearly, ${Y}$ is semimartingale, but may not be a Levy process.

Consider a Levy-type stochastic integral of the form (3). Let ${Y_c}$ be the continuous part of ${Y}$ defined by

$\displaystyle Y_c^i(t) = \int_0^t G^i(s) ds + \sum_j \int_0^t F_j^i(s) d W^j(s).$

Theorem 1 (Ito’s theorem) For each ${f\in C^2(\mathbb{R}^d)}$, ${t\ge 0}$, with probability 1,

$\displaystyle \begin{array}{ll} f(Y(t)) - f(Y(0)) = & \displaystyle \sum_i \int_0^t \partial_i f(Y(s-)) d Y_c^i(s) + \frac 1 2 \sum_{i,j} \int_0^t \partial_i \partial_j f(Y(s-)) d [Y_c^i, Y_c^j] (s) \\ & \displaystyle \quad + \int_0^t \int_{|x|\ge 1} [f(Y(s-) + K(s,x)) - f(Y(s-))] N(ds, dx) \\ & \quad \displaystyle + \int_0^t \int_{|x|<1} [f(Y(s-)+ H(s,x)) - f(Y(s-))] \tilde N(ds, dx) \\ & \quad + \displaystyle \int_0^t \int_{|x|<1} [f(Y(s-)+ H(s,x)) - f(Y(s-)) \\ & \quad \hspace{1in} \displaystyle - \sum_i H^i(s,x) \partial_i f(Y(s-))] \nu(dx) ds. \end{array}$

Another representation of above Ito formula is that, if ${Y}$ is of (3), then for each ${f\in C^2(\mathbb{R}^d), t\ge 0}$, with probability 1, we have

$\displaystyle \begin{array}{ll} f(Y(t)) - f(Y(0)) = & \displaystyle \int_0^t \sum_i \partial_i f(Y(s-)) d Y^i(s) + \frac 1 2 \sum_{ij} \partial_i \partial_j f(Y(s-)) d[Y_c^i, Y_c^j](s) \\ & \displaystyle + \sum_{0\le s \le t} [f(Y(s)) - f(Y(s-)) - \sum_i \Delta Y^i(s) \partial_i f(Y(s-))], \end{array}$

Note that a special case of Ito’s formula yields the following classical chain rule for differentiable functions ${f}$, when the process ${Y}$ is of finite variation:

$\displaystyle \begin{array}{ll} f(Y(t)) - f(Y(0)) = & \int_0^t \sum_i \partial_i f(Y(s-)) d Y^i(s) \\ & + \sum_{0\le s \le t} [f(Y(s)) - f(Y(s-)) - \sum_i \Delta Y^i(s) \partial_i f(Y(s-))]. \end{array}$

Here is famous Levy characterization of Brownian motion.

Theorem 2 (Levy’s characterization) Let ${M}$ be a continuous centered martingale, which is adapted to a given filtration ${(\mathcal{F}_t, t\ge 0)}$. If ${[M_i, M_j](t) = A_{ij} t}$ for each ${t\ge 0, }$ ${1\le i, j \le d}$ where ${A = (A_{ij})}$ is a positive definite symmetric matrix, then ${M}$ is an ${\mathcal{F}_t}$-adapted Brownian motion with covariance ${A}$.

Proof: Fix ${u\in \mathbb{R}^d}$ and define process ${Y_u(t) = e^{i(u, M(t))}}$, then by Ito’s formula, we obtain

$\displaystyle \begin{array}{ll} d Y_u(t)& = Y_u(t) d (i \sum_j u^j M^j(t)) + \frac 1 2 Y_u(t) d [ i \sum_j u^j M^j(t), i \sum_j u^j M^j(t)] \\ & = i Y_u(t) \sum_j u^j dM^j(t) - \frac 1 2 Y_u(t) \sum_{i,j} u^i u^j A_{ij} dt \end{array}$

By taking integral, and expectation

$\displaystyle \mathbb{E} (Y_u(t)) = 1 - \frac 1 2 (u, Au) \int_0^t \mathbb{E}(Y_u(s) ) ds.$

Hence, ${\mathbb{E} (Y_u(t)) = \exp\{- \frac 1 2 (u, Au) t\}}$. $\Box$

Now, We define the quadratic variation to the more general case of Levy-type stochastic integrals ${Y = (Y(t), t\ge 0)}$ of the form (3). For each ${t\ge 0}$, we define a ${d\times d}$ matrix-valued adapted process ${[Y, Y]}$ by the following prescription for its ${(i,j)}$th entry

$\displaystyle [Y^i, Y^j](t) = [Y^i_c, Y^j_c](t) + \sum_{0\le s \le t} \Delta Y^i(s) \Delta Y^j(s).$

Each ${[Y^i, Y^j](t)}$ is almost surely finite, and we have

$\displaystyle \begin{array}{ll} [Y^i, Y^j](t) = & \displaystyle \sum_{k=1}^m \int_0^T F_k^i(s) F_k^j(s) ds + \int_0^t \int_{|x|<1} H^i(s,x) H^j(s,x) N(ds, dx) \\ & \hspace{2in} \displaystyle+ \int_0^t \int_{|x|\ge 1} K^i(s,x) K^j(s,x) N(ds, dx). \end{array}$

The term ${d[Y^1, Y^2](t)}$, which sometimes called an Ito correction, arises as a result of the following formal product relations between differentials:

$\displaystyle d W^i(t) dW^j(t) = \delta^{ij} dt; \quad N(dt, dx) N(dt, dy) = N(dt, dx) \delta(x-y).$

Theorem 3 (Ito’s product formula) If ${Y^1}$ and ${Y^2}$ are real-valued Levy-type stochastic integrals of the form (3). Then for all ${t\ge 0}$, we have that

$\displaystyle Y^1(t) Y^2(t) = Y^1(0) Y^2(0) + \int_0^t Y^1(s-) dY^2(s) + \int_0^t Y^2(s-) d Y^1(s) + [Y^1, Y^2](t).$

For completeness, we will give another characterization of quadratic variation which is sometimes quite useful. If ${X}$ and ${Y}$ are real-valued Levy-type stochastic integrals of the form (3), then for each ${t\ge 0}$, w.p.1, we have

$\displaystyle [X,Y](t) = \lim^P_{n\rightarrow \infty} \sum_{j=0}^{n} (X(t_{j+1}^{(n)}) - X(t_{j}^{(n)})) (Y(t_{j+1}^{(n)}) - Y(t_{j}^{(n)}))$